First of all you have to select the callenge "Well Done" before starting the endurance mode.
Ok, now what you have to do is to kill the animals with fire, meaning that fire arrows can't hit them directly. That's because the kill will count as an arrow kill, not a kill by fire.
Just shoot a fire arrow on the ground near the small animal you aiming for. It will spread fire all around, most of the times killing the animal. It complicates things a little bit, but it's faster than crafting molotovs or look for lamps.
Don't forget to use the card for unlimited fire arrows to easy things up.